Nonlinear parabolic equations and systems arise in various applications, namely, physics, biology, financial mathematics and others. We consider stochastic differential equations to describe Markov processes associated with classical  and viscosity solutions of the Cauchy problem for some classes of nonlinear PDEs and derive probabilistic representations of these solutions. To deal with classical solutions of PDEs we need an extension of the classical SDE theory while to deal with viscosity solutions we need an extension of the forward-backward SDE theory. Finally, we give a description of numerical schemes to construct approximations to the PDE solutions based on the derived probabilistic  representations and consider the possibility to apply neural network based algorithms to solve approximately PDE problems in large dimensions. The results will be illustrated with some examples.